Noncommuting random products

HARRY FURSTENBERG (i) Introduction. Let Xy, X2, ■■-, X", ---be a sequence of independent real valued random variables with a common distribution function F (x), and consider the sums Xy + X2 + ■■• + X". A fundamental theorem of classical probability theory is the strong law of large numbers which asserts that with probability one, Xy + X2+ ■■■ + Xn ~ n xdF (x), provided that J"| x\ dF (x) is finite. It is natural to inquire whether there exist laws governing the asymptotic behavior of products X"X"-y ---Xy, where the X¡ are now identically distributed independent random variables with values in an arbitrary group. This type of question arises, for example, in considering solutions to differential or difference equations with random coefficients 13. To illustrate this, consider the problem of determining the asymptotic behavior of a random sequence <!;" satisfying £n = ut&n-i + c»£i!-2j where (u", v") forms a sequence of independent identically distributed random vectors. In this case we can write ßn + 11 -XX X fi1] x -h+1 VJ+1 so the rate of growth of the £" is governed by the behavior of the matrix product X"Xnᵧ--Xy. Bellman, who apparently was the first to consider questions of this kind, studied random products of 2 x 2 matrices with strictly positive entries 2. He showed that in a certain sense the weak law of large numbers holds for the entries of the matrices X"X"ᵧ -Xy. In 9 it was shown that a strong law of large numbers is valid. More precisely, if y is the typical entry of XnX" ᵧ--Xy, then for a certain constant a, n ~1 log y -* a with probability 1, provided the entries of X¡ are positive and bounded away from oo and 0 in an appropriate sense. It should be emphasized that for matrices with arbitrary entries, this type of result does not hold and the analysis of 9 breaks down. In the present investigation we shall consider a noncompact semi-simple Lie group G and independent G-valued random variables Xn with a common distribution p on G. The law of large numbers here can be given the following form. We shall exhibit a finite dimensional linear space "VG of functions \¡/ (g)